Null Space Calculator
Calculate the null space (kernel) of a matrix using Gaussian elimination.
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Null Space (Kernel)
The null space of a matrix A is the set of all vectors x such that Ax = 0:
\[ \text{null}(A) = \{x \in \mathbb{R}^n \mid Ax = 0\} \]
Properties:
- null(A) is a subspace of \(\mathbb{R}^n\)
- dim(null(A)) = n - rank(A)
- null(A) = {0} if and only if A is invertible
- null(A) = null(A^TA)
- null(A) ∩ range(A^T) = {0}
Methods to Find Null Space:
- Gaussian Elimination (Row Echelon Form)
- Singular Value Decomposition (SVD)
- QR Decomposition
Applications:
- Solving homogeneous systems
- Finding eigenvectors
- Image compression
- Signal processing
- Machine learning
Geometric Interpretation:
- Set of vectors mapped to zero
- Dimension of solution space
- Related to matrix rank
- Used in linear transformations